Another H-super magic decompositions of the lexicographic product of graphs

Abstract

Let H and G be two simple graphs. The concept of an H-magic decomposition of G arises from the combination between graph decomposition and graph labeling. A decomposition of a graph G into isomorphic copies of a graph H is H-magic if there is a bijection f : V(G) ∪ E(G) → {1, 2, ..., ∣V(G) ∪ E(G)∣} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant. A lexicographic product of two graphs G₁ and G₂,  denoted by G₁[G₂],  is a graph which arises from G₁ by replacing each vertex of G₁ by a copy of the G₂ and each edge of G₁ by all edges of the complete bipartite graph K_n, n where n is the order of G₂. In this paper we provide a sufficient condition for $\overline{C_{n}}[\overline{K_{m}}]$ in order to have a $P_{t}[\overline{K_{m}}]$-magic decompositions, where n > 3, m > 1,  and t = 3, 4, n − 2.